查看“︁Tanc函数”︁的源代码

'''Tanc 函数''' 定义如下<ref>Weisstein, Eric W. "Tanc Function." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/TancFunction.html {{Wayback|url=http://mathworld.wolfram.com/TancFunction.html |date=20150325182410 }}</ref>

: <math>\operatorname{Tanc}(z)=\frac {\tan(z) }{z}</math>
[[File:Tanc 2D plot.png|thumb|Tanc 2D plot]]
[[File:Tanc'(z) 2D plot.png|thumb|Tanc'(z) 2D plot]]
[[File:Tanc integral.png|thumb|Tanc integral 2D plot]]
[[File:Tanc integral 3D plot.png|thumb|Tanc integral 3D plot]]
;虚域虚部
*<math> \operatorname{Im} \left( \frac {\tan(x+iy) }{x+iy} \right) </math>
;虚域实部
*<math> \operatorname{Re} \left( \frac {\tan \left( x+iy \right) }{x+iy} \right) </math>
;绝对值
*<math>  \left| \frac {\tan(x+iy) }{x+iy} \right| </math>
;一阶导数
*<math> \frac {1- \tan(z))^2}{z} - \frac {\tan(z)}{z^2} </math>
;导数实部
*<math> -\operatorname{Re} \left( -\frac {1- (\tan(x+iy))^2}{x+iy} +\frac{\tan(x+iy)}{(x+iy)^2} \right)
    </math>
;导数虚部
*<math>-\operatorname{Im} \left( -\frac {1-(\tan(x+iy))^2}{x+iy} + \frac {\tan(x+iy)}{(x+iy)^2} \right) 
     </math>
;导数绝对值
*<math>  \left| -\frac{1-(\tan(x+iy))^2}{x+iy}+\frac {\tan(x+iy)}{(x+iy)^2} \right| </math>

==与其他特殊函数的关系==

* <math>\operatorname{Tanc}(z)={\frac {2\,i{{\rm KummerM}\left(1,\,2,\,2\,iz\right)}}{ \left( 2\,z+\pi 
 \right) {{\rm KummerM}\left(1,\,2,\,i \left( 2\,z+\pi  \right) \right)}}}</math>

*<math>\operatorname{Tanc}(z)={\frac {2\,i{\it HeunB} \left( 2,0,0,0,\sqrt {2}\sqrt {iz} \right) }{
 \left( 2\,z+\pi  \right) {\it HeunB} \left( 2,0,0,0,\sqrt {2}\sqrt {1
/2\,i \left( 2\,z+\pi  \right) } \right) }} </math>

* <math>\operatorname{Tanc}(z)={\frac {{{\rm WhittakerM}\left(0,\,1/2,\,2\,iz\right)}}{
{{\rm WhittakerM}\left(0,\,1/2,\,i \left( 2\,z+\pi  \right) \right)}z}}
 </math>

==级数展开==

: <math>\operatorname{Tanc} z \approx (1+{\frac {1}{3}}{z}^{2}+{\frac {2}{15}}{z}^{4}+{\frac {17}{315}}{z}^{6}+{\frac {62}{2835}}{z}^{8}+{\frac {1382}{155925}}{z}^{10}+{\frac {
21844}{6081075}}{z}^{12}+{\frac {929569}{638512875}}{z}^{14}+O \left( 
{z}^{16} \right) )</math>

<math>\int _{0}^{z}\!{\frac {\tan \left( x \right) }{x}}{dx}=(z+{\frac {1}{9
}}{z}^{3}+{\frac {2}{75}}{z}^{5}+{\frac {17}{2205}}{z}^{7}+{\frac {62}
{25515}}{z}^{9}+{\frac {1382}{1715175}}{z}^{11}+{\frac {21844}{
79053975}}{z}^{13}+{\frac {929569}{9577693125}}{z}^{15}+O \left( {z}^{
17} \right) )</math>


==图集==
{|
|[[File:Tanc abs complex 3D plot.png|thumb|Tanc abs complex 3D]]
|[[File:Tanc Im complex 3D plot.png|thumb|Tanc Im complex 3D plot]]
|[[File:Tanc Re complex 3D plot.png|thumb|Tanc Re complex 3D plot]]
|}
{|
|[[File:Tanc'(z) Im complex 3D plot.png|thumb|Tanc'(z) Im complex 3D plot]]
|[[File:Tanc'(z) Re complex 3D plot.png|thumb|Tanc'(z) Re complex 3D plot]]
|[[File:Tanc'(z) abs complex 3D plot.png|thumb|Tanc'(z) abs complex 3D plot]]
|
|}

{|
|[[File:Tanc abs plot.JPG|thumb|Tanc abs plot]]  
|[[File:Tanc Im plot.JPG|thumb|Tanc Im plot]]
|[[File:Tanc Re plot.JPG|thumb|Tanc Re plot]]
|}
{|
|[[File:Tanc'(z) Im plot.JPG|thumb|Tanc'(z) Im plot]]
|[[File:Tanc'(z) abs plot.JPG|thumb|Tanc'(z) abs plot]]
|[[File:Tanc'(z) Re plot.JPG|thumb|Tanc'(z) Re plot]]
|}
{|
|[[File:Tanc integral abs plot.png|thumb|Tanc integral abs plot]]
|[[File:Tanc integral Im plot.png|thumb|Tanc integral Im plot]]
|[[File:Tanc integral Re plot.png|thumb|Tanc integral Re plot]]
|}
{|
|[[File:Tanc abs complex 3D plot2.JPG|thumb|Tanc abs complex 3D plot]]
|[[File:Tanc Im complex 3D plot2.JPG|thumb|Tanc Im complex 3D plot]]
|[[File:Tanc Re complex 3D plot2.JPG|thumb|Tanc Re complex 3D plot]]
|}
==参看==
*[[Sinhc函数]]
*[[Coshc函数]]
*[[Tanhc函数]]
*[[双曲正弦积分函数]]

==参考文献==
<references/>

[[Category:特殊函数]]